Continuity properties of multi-parameter pseudodifferential operators on Bony class

Abstract

It is well-known that the pseudodifferential operator with the symbol in Bony class, a subset of \(S_{1,1}^0(\mathbb R^n)\), is bounded on \(L^{2}(\mathbb {R}^{n})\). The main purpose of this paper is to extend the classical results to multi-parameter case, i.e., to discuss the boundedness on \(L^2(\mathbb {R}^{n_1+n_2})\) and on \(h^{p}(\mathbb {R}^{n_{1}}\times \mathbb {R}^{n_{2}}) (0<p\le 1)\) of multi-parameter pseudodifferential operator with symbol satisfying multi-parameter Bony conditions.